Do the people that sold naked calls that expire in the money have to settle up at the closing price on the day of expiry?

A long call gives the owner the right to buy the stock at the strike price. As the seller, you are obligated to sell the underlying at the strike price.

If your short call expires in-the-money, you must deliver the shares if you own them. If you do not own them and if they are available, your broker will borrow them from another account and deliver them to the exerciser of the call. You will then be short the shares. If the shares are not borrowable, your broker will buy them for you on the open market (after hours) so that you can satisfy the assignment.

There is no involvement on your part with waiting for a better price after hours, or the next day before settling. This is all handled automatically by the OCC and your broker.